From London to Eltham: Trains run from Charing Cross, Waterloo East, London Bridge, Cannon Street and Victoria. You need the ones that say "via Bexleyheath". There are 2 stations that you can use - Eltham and Falconwood, depending on which end of Eltham you want.

From London to Mottingham and New Eltham: Trains run from Charing Cross, Waterloo East, London Bridge, Cannon Street and Victoria. You need the ones that say "via Sidcup".